Principles of Modern Radar - Volume 2 1891121537

9.9 Problems 451[32] Golub, G.H. and Van Loan, C.F., Matrix Computations, 3d ed., Johns Hopkins UniversityPress, 1996.[33] Anderson, E., et al., LAPACK Users’ Guide, 3d ed., Society for Industrial and Applied Mathematics,Philadelphia, PA, 1999.[34] Rader, C.M., ”VLSI Systolic Arrays for Adaptive Nulling,” IEEE Signal Processing Magazine,July 1996.[35] Farina, A., Antenna-Based Signal Processing Techniques for Radar Systems, Artech House,Boston, MA, 1992.[36] Special Issue on Adaptive Antennas, IEEE Transactions on AP, vol. 24, September 1976.[37] Weiss, s., et al., “An Efficient Scheme for Broadband Adaptive Beamforming,” in AsilomarConf. Sig. Sys. Comp., Monterey, CA, November 1999.[38] Vaidyanathan, P.P. “Multirate Digital Filters, Filter Banks, Polyphase Networks, and Applications:A Tutorial,” Proceedings of the IEEE, vol. 78, no. 1, pp. 56–93, January 1990.[39] Fliege, N.J., Multirate Digital Signal Processing: Multirate Systems—Filter Banks—Wavelets,Wiley, New York, 1994.[40] Aalfs, D.D., Brown, G.C., Holder, E.J., and Beason, B., “Adaptive Beamforming for WidebandRadars,” in Proceedings of the International Radar Symposium, Munich, Germany, pp. 983–992, September 15–27, 1998.9.9 PROBLEMS1. What is the equation for the availability of an N-channel DBF system that can toleratethe failure of any two receivers? What is the general form of the equation foravailability where P is the number of failed receivers that can be tolerated?2. The (3:1) overlapped subarray layout for the 64-element linear array used to generateFigures 9-17 and 9-18 has six subarrays instead of eight. Why?3. How could the example array from problem 2 be changed so that it will have eight3:1 overlapped subarrays?4. Derive the Wiener filter solution for the optimum adaptive weights by minimizing themean squared error of the beamforming output.5. Derive the maximum SINR solution for the optimum adaptive weights.6. Derive the LCMV solution using the method of Lagrange multipliers.7. According to the principle of orthogonality, the output, y(n), of the optimal filter isorthogonal to the error (i.e., E[y(n)e*(n)] = 0). Use this relationship to derive theoptimal solution for the weights.8. Write an expression for the covariance matrix of a plane wave in the presence ofnoise that is independent and identically distributed. The expression should be interms of the steering vector for the plane wave signal, the signal power, and the noisepower.9. Using the matrix inversion lemma, write an expression for the inverse of the covariancematrix from problem 6.(Matrix inversion lemma: a matrix of the form A = B −1 + CD −1 C H has an inversegivenbyA −1 = B + BC(D + C H BC) −1 C H B).